pieter belmans pieter.belmans@uantwerp · 2020. 7. 19. · relating db(kq) to k(zq) theorem...
TRANSCRIPT
Understanding Db(kQ) using moduli spaces
Pieter [email protected]
Goals
Bernhard Keller and Sarah Scherotzke, Graded quiver varieties andderived categories, arXiv:1303.2318v2:
1. connect Db(kQ) to a moduli variety M0(w);
2. describe the moduli variety in terms of Db(kQ) and vice versa.
My feeble goal:
3. generalise Db(kQ) to a derived moduli stack RPerfQ ;
4. describe the derived moduli stack using the moduli variety.
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Context
It generalises
1. Hiraku Nakajima, Quiver varieties and finite-dimensionalrepresentations of quantum affine algebras,arXiv:math/9912158
2. Hiraku Nakajima, Quiver varieties and cluster algebras,arXiv:0905.0002v5
3. Yoshiyuki Kimura and Fan Qin, Graded quiver varieties,quantum cluster algebras and dual canonical bases,arXiv:1205.2066v2
4. Bernard Leclerc and Pierre-Guy Plamondon, Nakajima varietiesand repetitive algebras, 1208.3910v2
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Conventions
1. k algebraically closed
2. Q a finite acyclic quiver
3. take Q connected for ease of statements
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A reminder on derived categories
Construction
1. A a ring;
2. Mod-A abelian category of A-modules;
3. Ch(Mod-A) abelian category of chain complexes of A-modules;
(4.) K(Mod-A) triangulated category of chain complexes up tohomotopy;
5. D(Mod-A) triangulated category of chain complexes withquasi-isomorphisms inverted.
Motivation
Natural location to do homological algebra.
5/30
A reminder on moduli spaces
Philosophy
A moduli “space” is an geometric object parametrising “families ofobjects”.
I A “space” could be: topological space, manifold, variety,scheme, stack, derived stack, . . .
I A “family of objects” could be: curves, algebra structures,modules, sheaves, subvarieties in a given variety, . . . Then thegeometric structure of the space determines which objects“look a like”.
1. moduli space of curves (= Riemann surfaces) Mg ,dimMg = 3g − 3
2. moduli space of algebra structures on finite-dimensionalvectorspace Algr 6/30
Repetition quivers
We need a (technical) construction. . .
Definition
The repetition quiver ZQ has as vertices
Q0 × Z = {(i , p) | i ∈ Q0, p ∈ Z}
and edges⋃α : i→j
{(α, p) : (i , p)→ (j , p);σ(α, p) : (j , p − 1)→ (i , p)} .
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Translations in repetition quivers
1. in the definition: σ : ZQ1 → ZQ1;
2. translation to the left: τ , both on ZQ0 and ZQ1;
3. we have σ2 = τ .
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Examples of repetition quivers
A3
0
1
2τ(2) τ−1(2)
βτ(β)
ZA3 . . . . . .
σ(β)
D4
0
3
2
1τ(1) τ−1(1)
βτ(β)
ZD4 . . . . . .σ(β)
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Framed quivers
Definition
The framed quiver Q of Q has vertices Q0 and Q ′0 = {i ′ | i ∈ Q0},
and edges Q1 and {i → i ′ | i ∈ Q0}. The vertices i ′ are the frozenvertices.
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Examples of framed quivers
A3
0
1
2
A3
0’
1’
2’
D4
0
3
2
1
D4
0’
1’
2’
3’
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Mesh categories
Definition
The mesh category k(ZQ) is the k-linear category withObj(k(ZQ)) = ZQ0 and
Homk(ZQ)(a, b) = 〈paths from a to b in ZQ〉/(urxv | x ∈ ZQ0)
where rx is the mesh relator associated to x , given by
rx =∑
β : y→x
σ(β)β :
y1
τ(x)... x
yn
β1σ(β1)
σ(βn) βn
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Remarks on mesh categories
This construction finds its origins in Auslander-Reiten theory.
Example
In the mesh category k(A2) all paths of length 2 or more areidentified with 0.
More interesting examples: see next, when we’ve introducedNakajima categories.
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Regular Nakajima categories
Definition
The regular Nakajima category RQ (or just R) is the mesh categoryon the framed quiver, where we only impose the mesh relators onthe non-frozen vertices.
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Examples of regular Nakajima categories
. . .RA3. . .
. . .RD4. . .
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Singular Nakajima categories
Definition
The singular Nakajima category SQ (or just S) is the fullsubcategory of the regular Nakajima category R on the frozenvertices.
We have regular versus singular because of the related modulivarieties: one is regular, the other can be singular.
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Examples of singular Nakajima categories
These categories become really hard to draw, see Keller–Scherotzkefor the case D4 which is next to impossible to reproduce.The singular Nakajima category for A2:
b
a
a
b
a
a
b
a
a
b
a
a
b
a
a
c c c c
a
a
b b
a a a
b c
with relations
1. ab − ba
2. a3 − cb
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Graded affine quiver varieties
Definition
The graded affine quiver variety M0(w) for a finitely supporteddimension vector w : Obj(S)→ N is the variety of S-modules M,such that M(x) ∼= kw(x).
M0(w) ∼=∏
x ,y∈Obj(S)
Homk
(HomS(x , y), kw(x)w(y)
)/I
where I is an ideal of relations: a module M is described by
1. images of the morphisms in S;
2. relations that hold in S.
Hence, M0(w) Zariski closed subset of an affine space!
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Structure of M0(w)
Understanding structure of S implies understanding M0(w). Wecan describe the quiver of S, with nodes Zσ(Q0).
Theorem (Keller–Scherotzke, 2013)
We have
#{σ(y)→ σ(x)} = dim Ext1S(Sσ(x),Sσ(y))
and
#{relations for σ(y) to σ(x)} = dim Ext2S(Sσ(x), Sσ(y)).
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Relating Db(kQ) to k(ZQ)
Theorem (Happel, 1987)
There exists a canonical fully faithful functor
H: k(ZQ)→ ind(Db(kQ))
such that the vertex (i , 0) is sent to the indecomposable projectivemodule Pi , for i ∈ Q0.It is moreover an equivalence if and only if Q is a Dynkin quiver.
Hence we get a relationship between the repetition quiver and thederived category!
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An isomorphism of Ext’s
Theorem
Let p ≥ 1. For all x , y ∈ ZQ0 we have
ExtpS(Sσ(x),Sσ(y)) ∼= HomDb(kQ)(H(x),Σp H(x)).
Moreover, if Q is not Dynkin these are zero for p ≥ 2.
Applying Keller–Scherotzke’s result:
Corollary
For Q not Dynkin there are no relations! We have M0(w)isomorphic to affine space.
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Stability and costability
Definition
An R-module is stable if for all x ∈ ZQ0 non-frozen we have
HomR(Sx ,M) = 0.
Interpretation
M does not contain a non-zero submodule supported only onnon-frozen vertices.
Dual definition for costable: HomR(M, Sx) = 0.
Interpretation
M does not have a non-zero quotient supported only on non-frozenvertices.
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Dimension vectors
We’ll denote (v ,w)
v : Obj(R) \ Obj(S)→ Nw : Obj(S)→ N
dimension vectors for the regular Nakajima category.
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A related moduli variety
Definition
The variety M(v ,w) is a moduli space for the R-modules M suchthat
1. M is stable;
2. M(x) ∼= kv(x);
3. M(σ(x)) ∼= kw(σ(x)).
There is moreover a (free) base change action by the group
Gv :=∏
x∈Obj(R)\Obj(S)
GLv(x)(k)
Only on the non-frozen vertices!
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Graded quiver varieties
Definition
The graded quiver variety M(v ,w) is the quotient M(v ,w)/Gv .
Using GIT this becomes a smooth quasi-projective variety, and therestriction res : Mod-R → Mod-S becomes a projection map
π : M(v ,w)→M0(w)
which is proper (“=” inverse images of compacts are compact).
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Stratification
Goal
A stratification of M0(w).
Definition
Denote by Mreg(v ,w) the open subset of M(v ,w) formed byisomorphism classes of R-modules which are also costable.
By varying the vector v (w is fixed) we can stratify M0(w) by theimages of the non-empty Mreg(v ,w), and each of these isisomorphic to its image in M0(w).
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Statement
Theorem (Keller–Scherotzke, 2013)
There is a canonical δ-functor
Φ: mod-S → Db(kQ)
such that
1. the simple module Sσ(x) for x ∈ ZQ0 is sent to H(x);
2. M1,M2 ∈M0(w) lie in the same stratum if and onlyif Φ(M1) ∼= Φ(M2) in Db(kQ).
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Applications
1. generalising the following result: Desingularization of quiverGrassmannians for Dynkin quivers, Giovanni Cerulli Irelli,Evgeny Feigin and Markus Reineke, arXiv:1209.3960
2. link with derived algebraic geometry and moduli spaces ofderived categories: Moduli of objects in dg categories, BertrandToen and Michel Vaquie, arXiv:math/0503269
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Derived moduli stacks
i (⊔
w M0(w)) RPerfS
RPerfQ
All of these objects are “derived”.
Questions
1. What are the geometric properties of these morphisms?
2. Do we obtain a smooth atlas for the moduli stacks?
3. Can we strengthen the results on the stratification?
4. Do these stacks have interesting intrinsic structure?
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Corollary in NCAG
Claim
The derived moduli stack of vector bundles on a noncommutativecurve is [−1, 0]-truncated, just like the commutative case.
Context
1. non-derived moduli stack VectC of vector bundles on acommutative curve C is smooth (no need for derivedness);
2. non-derived moduli stack VectS of vector bundles on acommutative surface S is singular (but derived smooth);
3. derived moduli stack of vector bundles (associated to Qnon-Dynkin) on a noncommutative curve is as nice as thecommutative counterpart.
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